In a class of $10$, there are $3$ students who are secretly robots. If the teacher chooses $2$ students, what is the probability that neither of them are secretly robots?
Answer: We can think about this problem as the probability of $2$ events happening. The first event is the teacher choosing one student who is not secretly a robot. The second event is the teacher choosing another student who is not secretly a robot, given that the teacher already chose someone who is not secretly a robot. The probabilty that the teacher will choose someone who is not secretly a robot is the number of students who are not secretly robots divided by the total number of students: $\dfrac{7} {10}$ Once the teacher's chosen one student, there are only $9$ left. There's also one fewer student who is not secretly a robot, since the teacher isn't going to pick the same student twice. So, the probability that the teacher picks a second student who also is not secretly a robot is $\dfrac{6} {9}$ So, the probability of the teacher picking $2$ students such that none of them are secretly robots is $\dfrac{7}{10}\cdot\dfrac{6}{9} = \dfrac{7}{15}$